Let two hypothesis be:
$H_0: \rm PMF(\mu)$ with prior $(1-p)$ and
$H_1: \rm PMF(\sigma)$ with prior $p$.
Is it true that the probability of total error $ \Big( P_e= \rm type \:I \: error + type \: II \: error \Big)$ is invariant to swapping the priors? i-e for $H_0$ with prior $p$ and $H_1$ with prior $(1-p)$.